Daniel Bernoulli's Principle Theorem

Bernoulli's theorem explores the behaviour of the ideal fluid flowing through the pipe. This important principle in fluid mechanics is found by Daniel Bernoulli in 1738

Daniel Bernoulli's Principle Theorem

STATEMENT
  Bernoulli's theorem states that when a liquid is flowing, the total of the pressure energy, kinetic energy and potential energy per unit mass should be constant.

EQUATION OF BERNOULLI'S THEOREM
The equation is
++gh=constant.

ASSUMPTION
Let us assume a streamline flow of liquid which has a density .It passes through a pipe XY. Assume that the pressures are denoted as P1 and P2. The cross sectional areas of the pipe are a1 and a2. These cross sectional areas are at the point X and Y. Let v1 be the velocity of the liquid which enters the pipe through X and v2 be the velocity of the liquid which leaves the pipe through Y. In this case when the height of the Y is greater than X, the liquid is accelerated. This acceleration should be forced against the force of gravity. Hence the pressure P1 is greater than the pressure P2

DERIVATION OF BERNOULLI PRINCIPLE:
Consider the mass m which is to be passed into the pipe, then
a1v1=a2v2=m

Dividing the equation by we get,
a1v1=a2v2==V

Force acting on liquid at X = p1a1

Force acting on the liquid at Y = p2a2

Work done per second at X = p1a1X v1=p1V

Work done per second at Y= p2a2XW2=p2V

Total work performed per second on the liquid by the pressure starting from X to Y =p1V-p2V

If the mass of the liquid is m, then the potential energy per second on the liquid from X to Y is equal to mgh2-mgh1

Now the increase of kinetic energy per second=mv22-mv12

The work energy principle states that the work done per second by the pressure is equal to adding the increase in potential energy per second with the increase in kinetic energy per second.

P1V-P2V=(mgh2-mgh1)+

P1V+mgh1+mv12= P2V+mgh2+mv22

On dividing by m throughtout,
+gh1+v12=+gh2+v22

+gh1+v12= +gh2+v22

+gh+v2=constant

When dividing by g, the term becomes

++h=constant

Here the terms ,,h are called pressure head,velocity head and gravitational head respectively.

Hence the Daniel Bernoulli's principle theorem is proved..
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